Copy local rotation on one axis of A to B

You could do it by taking Hadamard Product of desired axis of rotation for the plane and Euler angle component of the cube.Doing that will eliminate the values in all other axes than the desired one .And then update the plane's rotation with the newly obtained vector.

 v1 = (x, y, z)
 v2 =transform.forward =(0, 0, 1)
 Hadamard product of v1 and v2 gives (0, 0, z)

I have done the following implementation.Take a look and I hope it helps.

 public class HandleRotation : MonoBehaviour
 {
     public Transform _CubeTransform;
     public Transform _QuadTransform;
 
     
     void Update () {
         //Controllig the Cube's rotation with keyboard
         if(Input.GetKey(KeyCode.A))
             _CubeTransform.Rotate(_CubeTransform.forward,Space.Self);
         if(Input.GetKey(KeyCode.D))
             _CubeTransform.Rotate(_CubeTransform.forward*-1,Space.Self);        
         if(Input.GetKey(KeyCode.W))
             _CubeTransform.Rotate(_CubeTransform.up,Space.Self);
         if(Input.GetKey(KeyCode.S))
             _CubeTransform.Rotate(_CubeTransform.up*-1,Space.Self);
         //Update quad's rotation in the desired axis (Vector3.forward here )
         UpdateRotation(_QuadTransform,_QuadTransform.forward);
         print(transform.forward);
     }
         
     private void UpdateRotation(Transform target,Vector3 axis)
     {
         Vector3 rot = _CubeTransform.rotation.eulerAngles;
         rot = Hadamard(rot,axis);
         target.rotation = Quaternion.Euler(rot);
     }
 
     //returns hadamard product of two vectors
     private Vector3 Hadamard(Vector3 v1, Vector3 v2)
     {
         return new Vector3(v1.x*v2.x,v1.y*v2.y,v1.z*v2.z);
     }
 }